You cannot make her go to sleep. Say good night and leave the room. Be sure the room temperature is comfortable, she has her lovey, etc. Try a night light and/or soft music. It's Ok if she quietly plays or talks to herself while she's getting sleepy.
sounds like you're trying too hard to convince her that she should go to sleep. When you argue with her you are not only preventing her from getting sleepy but you're giving her the message that you know her body better than she does. she's at an age when she wants to be more independent. You can say to her that she may not be sleepy but it's night time and time to be in bed. You know she'll go to sleep when she's ready.
Sounds like you had a massive power struggle which, as you found, upsets everyone and doesn't get the job done. Just back off and let sleep happen once you've done the bed time routine and made the room comfortable.
Later: Not listening is typical of all children. That is why we make short statements while looking into their eyes. Sounds like you may be having a similar difficulty that I had when I first started parenting. I thought it was my job to explain things so that they would want to do what I'd asked. I started way too early asking for co-operation by explaining. Their minds aren't capable of grasping, let alone understanding, an explanation. Just tell her what you want her to do. At 3, if she doesn't do it, gently lead her into the start of doing it. Make as many things fun to do as possible. Play games with her. Don't expect her to do more than one thing at a time on her own.
Examples: When my grandchildren were that age, we raced to the car. We raced to get our seat belts fastened. We raced to get toys picked up. Counting to 3 or 5 or 10, depending on how long it will take her to do something, helps. Give her choices whenever possible. Do you want to wear this dress or this outfit? What color do you want to wear today?
And a really effective skill to use is patience, once they know that you won't give in. My granddaughter frequently had difficulty changing activities. I would tell her what we were going to do, give her time to put away the toys, or whatever, and then I'd just stand there waiting for her to move. After the first directions and getting the toys put away, don't say anything. Just quietly and patiently stand there looking at her. Most children get uncomfortable, just as do most adults, when someone is waiting for them.
It's also typical for girls to request their Daddy at this age. This is the age during which they are "wired" to search out their Dad. You've probably noticed she's become a bit of a flirt, too, and not only with her Dad.
I think you'll find it helpful to read up on child development. There is good info on the Internet as well as many good books.